# Integration by Substitution

1. Nov 30, 2004

### bross7

I'm stuck on how to advance further on this problem and if anyone can point my in the right direction I would be greatly appreciative.

$$\int\frac{dx}{\sqrt{x(1-x)}}$$

The integral has to be solved using substitution, but we are required to use
$$u=\sqrt{x}$$

From this:
$$du=\frac{dx}{2\sqrt{x}}$$

But I am stuck on how to convert the remaining portion of the function in terms of du.
$$\int\frac{dx}{u\sqrt{1-x}}$$

2. Nov 30, 2004

### ShawnD

I gave it a try and couldn't get anywhere with it. Maple says the answer is arcsin(2x-1).

Is that exactly how the question was given?

3. Nov 30, 2004

### Justin Lazear

$$u = \sqrt{x}$$

so

$$u^2 = x$$

and

$$2du = \frac{dx}{\sqrt{x}}$$

First use the third equation, then use the second equation to get rid of any other instances of x that're left.

And Shawn is not correct in his solution.

--J

Last edited: Nov 30, 2004
4. Dec 1, 2004

Shaun's solution looks good to me, what do you propose the actual answer is Justin?

5. Dec 1, 2004

### spacetime

Complete the square within the square root in the denominator and the apply the result

$$\int\frac{dx}{\sqrt{a^2-x^2}} = arcsin\frac{x}{a}$$

spacetime
www.geocities.com/physics_all

6. Dec 1, 2004

### Justin Lazear

$$\int\frac{dx}{\sqrt{x(1-x)}} = 2 \arcsin{\left(\sqrt{x}\right)}$$

Differentiate it and you'll get the integrand.

The derivative of arcsin(2x-1) is $$\frac{2}{\sqrt{4x^2 - 4x + 2}}$$.

--J